NONLINEAR BEHAVIOUR OF LONG-SPAN CABLE-STAYED BRIDGES Domenico Bruno* Antonio Grimaldi** MECCANICA 20 (1985), 303-313 SOMMARIO. In questo lavoro viene analizzato il comporta- mento non lineare dei ponti strallati di grande luce sotto l'azione dei carichi accidentali. Nell'analisi vengono utilizzati due modelli strutturali; il primo, di tipo continuo, si basa sull'ipotest di uniforme distribuzione della strallatura lungo la travata, il secondo, di tipo discreto, tiene conto invece dell'effettiva spaziatura degli stralli. Vengono infine presen- taft i risultati numerici ottenuti. SUMMARY. In this paper the nonlinear behaviour of long- span cable-stayed bridges is analyzed. Two different struc- tural models are presented. The first is obtained by assuming a continuous distribution of the stays along the deck, the second one is a discrete model which accounts for the actual stays spacing. Some numerical results are presented. LIST OF SYMBOLS E Young modulus E* Dischinger's secant modulus E t Dischinger's tangent modulus. g dead load p design live load k live load parameter I girder inertia K towers stiffness e dimensionless girder stiffness parameter L central span length s lateral span length H tower height A stays spacing a dimensionless bridge parameter o o anchor stays stress cra allowable stress of the stays 8 midspan deflection o girder's vertical displacements w girder's horizontal displacement u pylon tops horizontal displacement 1. INTRODUCTION In the last years a great interest has been shown for the scheme of long-span cable-stayed bridges, particularly regard- ing the so called fan-shaped scheme of the self anchored type. [ 1 - 2 - 3 ] include a recent revue of the problems and * Dipartimentodi Strutture, Urdversit~t dellaCalabria,Italia. ** Facolt~ di Ingegneria, II Universit~ di Roma, Italia. advantages of this solutions and a report about the latest and most interesting realizations. The static behaviour of this scheme is marked by a distri- bution of axial tensions in the stays and of axial compression in the girder, while less important is the bending stress as a result of the prevailing truss behaviour of the scheme. The research methods usually refer to a linear elastic analysis and the extensional stiffness of the stays is evaluated by the Dischinger elastic modulus [1 - 3]. This analysis is fairly approximate for 'bridges with a central span which is as large as those at presentconstructed (about 500 m), but the same analysis is too approximate for bridges with span longer than 700-800 m. In that case a more precise analysis is required for the exact evaluation of some effects which are essential for the bridge feasibility and designing. For example, above all in the presence of a railway load, there is ~he problem of the bridge high deformability. An exact evaluation of the displacements requires a non- linear analysis. In this paper a numerical and analytical research is made by considering the main nonlinear effects; that is: - nonlinear behaviour of the single stay; - instability effect of the axial compression in the girder. The analysis is first developed by working on a continuous model, already used in previous researches [4 ], and obtained by assuming a continous distribution of the stays stiffness along the girder. It is shown how, through a perturbative solution techni- que, it is possible to obtain solutions which are approximate but capable to account for the main nonlinear effect that is the extensional effect of the stays. A simple expression is deduced of the maximum midspan deflection of the bridge. The deformations and stresses produced by a general live load are then numerically evaluat- ed by means of an iterative solution method of the equili- brium equations corresponding to a discrete model. By using the discrete model the girder stability has also been studied where uniform load is applied on the central span. The highly nonlinear behaviour of the stays in case of unloading produces the buckling of the lateral spans of the bridge and the maximum value of the load parameter is found. Previous contributions to the study of nonlinear beha- viour of the cable-stayed bridges are given in [5] where a general analysis of the stays nonlinearity is presented by using a variational formulation. An analysis of the geome- tric nonlinearities and of the nonlinear effects produced by the stays is also contained in [6]. 20 (1985) 303